Embeddings in Cp-spaces

We survey Cp-theory from the point of view provided by the following question: when can a topological space be embedded into Cp(X), where X belongs to a given class of spaces? The question turns out to have interesting links to many general and concrete situations. In particular, we discuss from this position t-, and l-equivalences, continuous extenders, a theorem of Grothendieck, and the topological structure of compact subspaces of Cp(X), where X is Lindelِf. Many open problems, arising naturally under this approach, are presented, some of them new. Some results in the paper are also formulated for the first time, in particular, Theorems 2.29, 2.30, 2.38, 3.24, 5.3, 5.24, 5.25 and Corollaries 2.28, 3.23.